Problem: All of the 5th grade teachers and students from Loyola went on a field trip to an art museum. Tickets were $$8.50$ each for teachers and $$4.00$ each for students, and the group paid $$58.00$ in total. The next month, the same group visited a science museum where the tickets cost $$25.50$ each for teachers and $$8.50$ each for students, and the group paid $$153.00$ in total. Find the number of teachers and students on the field trips.
Answer: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8.5x+4y = 58}$ ${25.5x+8.5y = 153}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-25.5x-12y = -174}$ ${25.5x+8.5y = 153}$ Add the top and bottom equations together. $ -3.5y = -21 $ $ y = \dfrac{-21}{-3.5}$ ${y = 6}$ Now that you know ${y = 6}$ , plug it back into $ {8.5x+4y = 58}$ to find $x$ ${8.5x + 4}{(6)}{= 58}$ $8.5x+24 = 58$ $8.5x = 34$ $x = \dfrac{34}{8.5}$ ${x = 4}$ You can also plug ${y = 6}$ into $ {25.5x+8.5y = 153}$ and get the same answer for $x$ ${25.5x + 8.5}{(6)}{= 153}$ ${x = 4}$ There were $4$ teachers and $6$ students on the field trips.